Subgroups — #31797824

Order 17536.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578500	C₂	1	1	yes	maximal, characteristic
2	#22578500	C₂	1	1	yes	maximal, characteristic
3	#22578410	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555522	—	2	1	no	—
5	C₂×#14555522	—	2	1	no	—
6	C₂×#14555520	C₄	1	1	yes	characteristic
7	C₂×#14555518	C₂²	1	1	yes	characteristic
8	#17984155	—	2	1	no	—
9	C₄⋊#11764047	—	2	1	no	—
10	#17984046	C₄	1	1	yes	characteristic
11	#14555522	—	4	1	no	—
12	#14555522	—	4	1	no	—
13	#14555520	—	4	1	no	—
14	#14555518	—	4	1	no	—
15	C₂×#11764047	—	2	1	no	—
16	C₂×#11764047	D₄	1	1	yes	characteristic
17	C₂×#11764047	D₄	1	1	yes	characteristic
18	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₄⋊Dic₁₃₇	C₈	1	1	yes	characteristic
20	#14555487	—	2	1	no	—
21	C₄×Dic₁₃₇	—	2	1	no	—
22	C₄⋊C₄×C₁₃₇	C₈	1	1	yes	characteristic
23	#11764047	—	2	1	no	—
24	#11764047	—	2	1	no	—
25	#11764047	—	4	1	no	—
26	C₂²×D₁₃₇	M₁₆	1	1	yes	characteristic
27	C₂×Dic₁₃₇	—	2	1	no	—
28	C₂×Dic₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₄×D₁₃₇	—	2	1	no	—
30	C₄×D₁₃₇	—	2	1	no	—
31	C₂×C₄×C₁₃₇	—	2	1	no	—
32	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
34	C₂×D₁₃₇	—	2	1	no	—
35	C₂×D₁₃₇	—	2	1	no	—
36	C₂×D₁₃₇	—	2	1	no	—
37	C₄×C₁₃₇	—	2	1	no	—
38	C₄×C₁₃₇	—	4	1	no	—
39	Dic₁₃₇	—	4	1	no	—
40	Dic₁₃₇	#99	1	1	yes	characteristic
41	Dic₁₃₇	#101	1	1	yes	characteristic
42	C₂×C₁₃₇	#324	1	1	yes	characteristic
43	C₂×C₁₃₇	#350	1	1	yes	characteristic
44	C₂×C₁₃₇	#350	1	1	yes	characteristic
45	D₁₃₇	—	4	1	no	—
46	C₁₃₇	#1323	1	1	yes	characteristic, complemented, Sylow
47	#1323	—	137	1	no	maximal, Sylow
48	#424	—	137	1	no	—
49	C₂²⋊C₁₆	—	137	1	no	—
50	C₂²⋊C₁₆	—	137	1	no	—
51	C₂×M₁₆	—	137	1	no	—
52	C₂²×C₈	—	137	1	no	—
53	#117	—	137	1	no	—
54	C₂×C₁₆	—	274	1	no	—
55	C₂×C₁₆	—	274	1	no	—
56	C₄⋊C₈	—	274	1	no	—
57	C₄⋊C₈	—	274	1	no	—
58	C₂²×C₄	—	137	1	no	—
59	M₁₆	—	548	1	no	—
60	C₂×C₈	—	274	1	no	—
61	C₂×C₈	—	548	1	no	—
62	C₂×C₈	—	137	1	no	—
63	C₂×C₈	—	137	1	no	—
64	C₄⋊C₄	—	137	1	no	—
65	C₄⋊C₄	#11764049	1	1	yes	characteristic
66	C₂²⋊C₄	—	274	1	no	—
67	C₄²	—	274	1	no	—
68	C₁₆	—	548	1	no	—
69	C₁₆	—	548	1	no	—
70	C₂³	—	137	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	274	1	no	—
73	C₂×C₄	—	274	1	no	—
74	C₂×C₄	—	274	1	no	—
75	C₂×C₄	—	137	1	no	—
76	C₂×C₄	C₂×#11764049	1	1	yes	characteristic
77	C₈	—	274	1	no	—
78	C₈	—	274	1	no	—
79	C₈	—	548	1	no	—
80	C₂²	—	274	1	no	—
81	C₂²	—	274	1	no	—
82	C₂²	—	274	1	no	—
83	C₂²	C₂²⋊#11764049	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	548	1	no	—
87	C₄	—	137	1	no	—
88	C₄	—	137	1	no	—
89	C₂	—	548	1	no	—
90	C₂	#22578489	1	1	yes	characteristic
91	C₂	#22578503	1	1	yes	characteristic
92	C₂	#22578504	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#11764049
commutator	C₂×C₄×C₁₃₇	C₂×C₈
Frattini	C₂×C₄	C₂×#11764049
Fitting	C₄⋊C₄×C₁₃₇	C₈
socle	C₂²×C₁₃₇	C₂²⋊C₈
prime radical	C₂²×D₁₃₇	M₁₆
cosocle	C₂×#14555518	C₂²
hypercenter	C₄⋊C₄	#11764049
hypocenter	C₁₃₇	#1323
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
46. C₁₃₇	#1323	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22578500	2
#22578410	2
#1323	137